ECONOMICS : MATHEMATICS FOR ECONOMISTS III
Yale University
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Mathematical Quiz And Mathematical solutions
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It seems you're requesting assistance with a problem set related to Game Theory and Economics (ECON 159). I'd be happy to help you with specific questions or problems from that set. Please provide the details of the problems or questions you need assistance with, and I'll do my best to provide guidance and solutions in English.





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GAME THEORY, ECON 159, PROBLEM SET 9
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GAME THEORY, ECON 159, PROBLEM SET 9 
 
(i) If the n firms sustain a cartel that charges the monopoly price of 60 in each period, then the cartel output in each period is: 
 
 Q=100 −pm=100 −60 =40.
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Practice Exam Game Theory 1
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Practice Exam Game Theory1 Consider the following game in normal form:
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TESTS OF CONVEXITY AND CONCAVITY USING HESSIAN DETERMINANTS : MATHEMATICS FOR ECONOMISTS III
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A concave function has an absolute maximum when plotted on a graph, while a convex function 
gives an absolute minimum when plotted on a graph. 
It is easy to check for concavity and convexity in functions of single independent variable by just 
checking the second order derivative. 
If the second order derivative is negative, the function is concave and if positive, the function is 
said to be convex. 
In functions of two independent variables, there is a corresponding test which also use...
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FUNCTIONAL DEPENDENCE AND SOLUTION TO SIMULATNEOUS EQUATIONS : MATHEMATICS FOR ECONOMISTS III
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The Jacobian matrix can help to test whether there exists functional linear dependence in a set of 
݊ functions in ݊ variables. It is a matrix of the first order partial derivatives. If the functions are 
linearly dependent, the determinant of the Jacobian matrix is 0.
Fear of missing out? Then don’t!
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SOLUTION OF LPP USING SIMPLEX METHOD : MATHEMATICS FOR ECONOMISTS III
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In our lecture two, we were able to see how a LPP can be solved using the graphical method. In 
this lecture we will learn how to a LPP can be solved using the simplex method. 
Steps 
• Formulate the problem in the standard way. 
• Convert the inequalities into equations through the addition of a slack variable if the 
inequality is of a less than type (≤). 
• A slack variable represents unused capacity in the constraint. 
 E.g. 4X1+6X2≤180 Labour hours constraint 
 4X1+6X2 + S1 ...
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GRAPHICAL SOLUTION TO LINEAR PROGRAMMING : MATHEMATICS FOR ECONOMICS III
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In our lecture one, we were able to see how a LPP can be formulated in a standard way. In this 
section we will see how a LPP can be solved using the graphical method. 
Graphical methods of solving Linear programming can only be used for problems with TWO 
unknowns or decision variables. Problems with THREE OR MORE unknowns must be solved 
by techniques such as the simplex method. Graphical methods are the simplest to use and should 
be used wherever possible. Graphical method can be used t...
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